. Differential and integral calculus, an introductory course for colleges and engineering schools. along that arc. (decreasing) 55. Convexity and Concavity. An arc of a curve y = f(x) is said to be \ [ when viewed from below (that is, when viewed concave\ At exceptional points it may be a right angle. 55 SOME GENERAL PROPERTIES OF FUNCTIONS 71 by an observer stationed at the foot of the page) when it has suchshape as the arc AB, BC, or AC of Fig. j >. Let a point Ptraverse one of these arcs from left to right, carrying with it a tan-. Y V \ vi p^ \ / A £ T| K / 0 X a b o Fig. 1. Fig. 2. gen
. Differential and integral calculus, an introductory course for colleges and engineering schools. along that arc. (decreasing) 55. Convexity and Concavity. An arc of a curve y = f(x) is said to be \ [ when viewed from below (that is, when viewed concave\ At exceptional points it may be a right angle. 55 SOME GENERAL PROPERTIES OF FUNCTIONS 71 by an observer stationed at the foot of the page) when it has suchshape as the arc AB, BC, or AC of Fig. j >. Let a point Ptraverse one of these arcs from left to right, carrying with it a tan-. Y V \ vi p^ \ / A £ T| K / 0 X a b o Fig. 1. Fig. 2. gent to the curve. From the figures it can be seen that, if the arcconvex be , as this tangent rolls along the curve, the angle (concave) i • i • i • , ^r • ii \ increases ) , which it makes with OX continually , and that (decreases) .-. x . (an increasing ).,.,„ ^ therefore f (x) is function along the arc. From f a decreasing ) the figures it can be seen that the converse of this is also true, viz., increasing ) . ( convex ) , . \ throughout an arc, that arc ^s decreasing ) ( concave ) when viewed from below. Applying corollary 1 of theorem 3, Art. 54, we have Theorem 4- Iff(%) is ] [at every point of an arc, that arc If f(x) is IS convexconcave j when viewed from below. (convex ) arcs of a curve, y = fix). (concave) u Jy n Hence, to determine the concave we have only to determine the intervals within which fix) or y is 72 DIFFERENTIAL CALCULUS §56
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